The invention relates to the field of peach trees. More specifically, the invention is a new and distinct cultivar of peach tree Prunus persica L. Batsch named ‘S 6359’. The new peach tree ‘S 6359’ is a product of the breeding program at Angers, France. ‘S 6359’ was one of several seedlings resulting from a cross of unknown parents. ‘S 6359’ was asexually propagated by budding at Angers, France, and has been observed to remain true to type over successive asexually propagated generations.